Large-Eccentricity Asymptotics and Fast Analytic Approximation for Fourier modes of Post-Newtonian Eccentric Waveforms

Abstract

In this work, we develop analytic asymptotic methods for computing the Fourier modes of gravitational waves from post-Newtonian binary systems in the quasi-Keplerian parametrization in the high eccentricity regime. We also derive the large-eccentricity asymptotic expansion of the eccentricity enhancement function appearing in the tail contributions to the radiation. Furthermore, based on these results, we construct an endpoint-constrained analytic approximation that significantly accelerates the computation of the Fourier modes at large eccentricity. The overall error of this analytic approximation is controlled within $10^{-3}$, and it remains valid for Fourier modes with $p\le200$ and $e\le 0.9$. This approach provides analytic building blocks for modeling frequency-domain gravitational waves from highly eccentric binaries.